{"id":"13454e59-a95e-4016-8e5b-137782f26fda","arxiv_id":"2412.00779","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For degenerate parabolic and elliptic equations in divergence form on the upper half space, the paper establishes well-posedness and weighted mixed-norm Sobolev estimates under partially BMO coefficients.","lead":"This paper proves that a broad family of equations whose leading coefficients vanish or blow up at the boundary of a half-space have unique solutions with Sobolev regularity in weighted spaces. The result gives a unified maximal-regularity framework for applications ranging from option pricing to degenerate elliptic geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorems rest on an unproved generalized nondegenerate H^1_{p,ω} estimate (Remark 3.7); if the reduction to [6] and [9] fails for a0=a0(x_d), Lemma 3.6 and the perturbation arguments collapse.","rationale":"The reader's weakest assumption identifies the same load-bearing point: Lemma 3.6 relies on a nondegenerate H^1_{p,ω} estimate that is only outlined in Remark 3.7. This is indeed the most critical unproved dependency in the paper. The reduction to [6] and [9] is plausible, but the manuscript does not supply the proof, and the a0(x_d) case requires a genuine change of variables that could affect the partial-BMO and A_p-weight hypotheses. Since this estimate is used to localize and control the higher-order terms in every main theorem, the conditional verdict is appropriate. The optimality claim about the θ-range is a separate assertion, explicitly deferred to future work, and does not undermine the well-posedness statements; it should be marked as a conjecture rather than a proved result. I found no internal contradiction that would require rejection; the main issue is missing proof detail for a key auxiliary theorem.","tokens_in":41491,"tokens_out":31173,"duration_ms":280445,"concrete_test":"Write out the a0=a0(x_d) case of Remark 3.7 completely: after the bi-Lipschitz change y_d=∫_0^{x_d} a0(s)ds, derive the H^1_{p,ω}(T) estimate for v_t - D_k(\\tilde a_{kl}D_l v)+λ c0(y_d)v = D_iG_i+g, where \\tilde a_{dd}=a_{dd} a0(y_d). Verify that \\tilde a_{kl} satisfies the small partial-BMO condition of [6, Assumption A/A'] and reproduce the mean-oscillation argument of [9, Prop. 4.1, Lemma 5.1, Thm 2.8] line by line to obtain (3.25). If any step requires differentiability or extra smallness of a0 beyond (1.3), or if the A_p time weight fails to behave well under the y-transformation, then (3.25), and hence Theorems 2.6, 2.9, and 2.14, are not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central well-posedness claims in Theorems 2.6, 2.9, and 2.14 depend on the higher-order a priori estimate Lemma 3.6. Its proof applies, at (3.25), an H^1_{p,ω}(T) solvability estimate for the nondegenerate equation a0 v_t - D_i(a_ij D_j v)+(λ+λbar)c0 v = D_iG_i+g, with a0 allowed to be merely bounded measurable in t or x_d. This auxiliary estimate is not proved in the paper: Remark 3.7 only sketches a reduction via the change y_d=∫_0^{x_d} a0(s)ds, division by a0, and then 'following the arguments in [6]' and 'repeating the arguments in [9]'. The manuscript itself flags this as an outline, not a proof. If the transformed coefficients fail to satisfy the small partial-BMO hypotheses of [6] with constants independent of a0, or if the A_p time weight interacts badly with the a0(x_d) transformation in the mean-oscillation argument, then (3.25) is unsupported. Since Lemma 3.6 is invoked in Theorem 4.5, Lemma 5.1, and the proofs of all main theorems, the well-posedness conclusions would lose their foundation. The separate optimality claim in the introduction is deferred to future work and should be labeled as a conjecture; it is not load-bearing for the main estimates.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies degenerate divergence-form parabolic and elliptic equations in the upper half-space R^d_+, with leading coefficients of the form x_d^2 a_ij, where a_ij are bounded, uniformly elliptic and measurably depend on (t,x_d), except a_dd which depends only on t or only on x_d, and have small partial BMO in the remaining spatial directions. The main results, Theorems 2.6, 2.9, and 2.14, establish well-posedness in weighted mixed-norm Sobolev spaces H^1_{q,p,theta,omega} together with estimates of the form (1+sqrt(lambda))||u|| + ||x_d D_x u|| <= N(||x_d^{-1} F|| + (1+sqrt(lambda))^{-1}||f||), either for large lambda under a general partial-BMO assumption or, when lambda=0, under stronger ratio assumptions with the admissible range alpha p < theta < beta p determined by the roots of the quadratic (2.8). The proof proceeds through zeroth-order energy estimates using weighted Hardy inequalities, higher-order localization to nondegenerate equations, level-set arguments with a crawling-of-ink-spots lemma, duality, and interpolation.","tokens_in":41829,"tokens_out":7539,"duration_ms":69840,"significance":"If the technical gaps are filled, this is a valuable contribution: it extends weighted mixed-norm L_p theory to a broad class of degenerate divergence-form operators with partially BMO coefficients, allows a_dd to be merely measurable in one variable, and identifies an explicit theta-range depending on the lower-order coefficient ratios, with connections to Black-Scholes-type equations and Loewner-Nirenberg-type problems. The zeroth-order estimates are derived from first principles with explicit test functions and Hardy inequalities, and the paper gives careful parameter dependencies. There is no circular derivation: the prior results cited are used as ingredients rather than as restatements of the target theorems. However, a load-bearing higher-order estimate is only outlined by reference to previous work, and the claimed optimality of the theta-range is deferred to future work.","major_comments":[{"comment":"The existence part of Lemma 4.1 invokes [4, Theorem 8.2] for local solvability on the sets A_k and then states, without proof, that the same result can be obtained for nonconstant a0 by following the proof of the theorem. The same phrase appears in the proof of Theorem 2.14. If this generalized local solvability is not established, the approximation argument producing u_k and the subsequential solution is incomplete. Please provide the details or state the precise cited theorem that already covers variable a0 and variable c0 in the needed form.","section":"§4, Lemma 4.1"}],"minor_comments":[{"comment":"In (4.21) the constant is written as N = N(d,p,q,theta,nu,K,K0,nu), with nu repeated; the same duplication appears in Lemma 5.1.","section":"Theorem 4.5"},{"comment":"Please cite precise theorem or lemma numbers in [6] and [9] rather than saying 'following the arguments' and 'repeating the arguments'; this would greatly help verification of the claimed reduction.","section":"Remark 3.7"},{"comment":"Reference [34] contains a typo: 'N.V. Krylov,,' has a double comma.","section":"References"},{"comment":"The definition of the maximal function contains a miswritten expression: 'sup_{t in (s-r,r)}' should be a supremum over intervals containing the point at which the maximal function is evaluated.","section":"Lemma 4.3"},{"comment":"The sentence 'except a_dd, which is measurable in t or x_d' would be clearer as 'except a_dd, which is measurable only in t, or only in x_d'.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' own earlier work; I see no circularity, but the unproved generalizations in Remark 3.7 and Lemma 4.1 should be scrutinized by the editor. The optimality claim is not proven and should be relabeled as a conjecture. The main estimates are plausible and the overall strategy is sound, but the missing higher-order estimate is load-bearing and should be supplied before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read of Dong–Ryu 2412.00779.\n\nThe headline result is genuinely new: weighted mixed-norm Sobolev well-posedness for L u = a0 u_t - x_d^2 D_i(a_ij D_j u)+lower terms with coefficients satisfying small partial-BMO in the tangential variables and measurability in (t,x_d) — with a_dd measurable in only one of them. That's a real jump from constant coefficients or uniformly continuous coefficients in [31,39,24]. The θ-range αp<θ<βp tied to the roots of z^2+(1+n_b+n_hatb)z-n_c=0 is explicit, and the analysis makes clear why it appears.\n\nThe proof is long but organized. The zeroth-order estimates (Lemma 3.4) are self-contained and detailed: test function, weighted Hardy, a change of variables to symmetrize the matrix. The mixed-norm step via level sets and the crawling-ink-spots lemma is also written out. Most of the structure is sound.\n\nThe soft spot is Lemma 3.6. The higher-order estimate applies an H^1_{p,ω} solvability result for a nondegenerate equation a0 u_t - D_i(a_ij D_j u)+λ c0 u = ... with a0 merely measurable in t or x_d. This is not proved in the paper. Remark 3.7 sketches a reduction: divide by a0, change y_d = ∫ a0, then follow [6] for ω=1 and [9] for weights. The reduction looks plausible — the new coefficients stay in the partially BMO class, with constants depending on K. But the details of the weighted mean-oscillation argument are not shown, and the citation to [9] (time-fractional equations) is not obviously on point. If that auxiliary estimate fails, the localization argument and hence Theorems 2.6/2.9/2.14 lose their footing. This is load-bearing, but it's also a known type of result; a referee can likely fill it or point to the correct reference.\n\nA second, minor issue: the introduction claims the θ-range is optimal, but proof is deferred to future work. The main theorems don't need that claim; it should be labeled as a conjecture. There are also small typos (e.g., ν listed twice in Theorem 4.5's constant), but nothing structural.\n\nWho should read this: anyone working on L_p theory for degenerate elliptic/parabolic equations or SPDEs with degenerate operators. It deserves a serious referee. My recommendation: send it to peer review with a request that the authors either prove the auxiliary nondegenerate estimate in an appendix or pinpoint a theorem that covers exactly the (a0,c0) generality, including the A_p weight. Conditional acceptance is appropriate.\n\nFor me: cite-worthy if the auxiliary estimate checks out; I'd bring it to reading group.","headline":"A substantial extension of degenerate divergence-form maximal regularity, with one imported nondegenerate estimate that is only sketched; worth a serious referee but expect to chase the reduction in Remark 3.7.","tokens_in":42329,"tokens_out":6242,"would_cite":true,"duration_ms":56026,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J70","35K65","35D30","35R05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves well-posedness in weighted mixed-norm Sobolev spaces for degenerate parabolic and elliptic equations in divergence form on the upper half-space with $x_d^2$-vanishing coefficients.","keywords":["degenerate linear equations","divergence form","existence and uniqueness","weighted Sobolev spaces","weighted mixed-norm estimates","partial BMO coefficients","Muckenhoupt weights","upper half space"],"falsifier":"A concrete check is to verify the unproved nondegenerate estimate from Remark 3.7 by testing $a_0(x_d) = 1 + \\varepsilon \\sin(x_d)$, $c_0 = 1$, $a_{ij} = \\delta_{ij}$ on the half-space: if the $H^1_{p,\\omega}$ a priori bound fails with constants independent of $\\varepsilon$ for some $p \\in (1,\\infty)$ and $A_p$ weight $\\omega$, then Lemma 3.6 and the main theorems lose their foundation; for the optimality claim at $\\lambda=0$ ($d \\geq 2$), one can test whether the estimate (2.11) holds for some $\\theta$ outside $\\alpha p < \\theta < \\beta p$ using constant-coefficient operators with non-radial forcing, since the one-dimensional explicit solution (6.4) identifies the two endpoints as the only singular exponents.","tokens_in":41316,"feed_emoji":"📐","tokens_out":19008,"duration_ms":143595,"temperature":0.7,"pith_summary":"This paper establishes well-posedness and explicit a priori estimates for a broad class of degenerate parabolic and elliptic equations in divergence form on the upper half-space, where the leading coefficients vanish like $x_d^2$ at the boundary. The coefficients are allowed to be rough: bounded, uniformly elliptic, measurable in the normal variable (and in time for the parabolic case), with only small mean oscillations in the tangential directions. Solutions are sought in weighted mixed-norm Sobolev spaces built on the integrability of $u$ and $x_d D_x u$, with a Muckenhoupt weight in time. A key feature is that when the zeroth-order damping $\\lambda c_0 u$ is removed, the admissible weight exponent $\\theta$ is tied to the ratios of the lower-order coefficients to $a_{dd}$ through an explicit quadratic equation, and the paper claims this range is optimal, deferring the proof to future work. If correct, the estimates provide maximal regularity for degenerate equations arising in finance, stochastic control, and conformal geometry.","feed_headline":"Degenerate PDEs solved in weighted Sobolev spaces","feed_subtitle":"Parabolic and elliptic equations with boundary-vanishing coefficients get unique solutions and explicit bounds.","key_machinery":"The argument rests on three devices. First, testing the equation against $|u|^{p-2}u\\,x_d^{\\theta-1}$ combined with weighted Hardy's inequality yields the zeroth-order estimate, aided by a change of variables $y_i = x_i - \\int_0^{x_d} (a_{id}+a_{di})/(2a_{dd})\\,dr$ that makes the leading coefficient diagonal. Second, a localization argument freezes coefficients on balls, applies an assumed $H^1_{p,\\omega}$ estimate for the nondegenerate frozen equation, and reassembles the pieces with a weighted partition of unity. Third, a level-set maximal-function argument ('crawling of ink spots') upgrades unmixed-norm estimates to mixed-norm estimates in time under Muckenhoupt weights. The quadratic $z^2 + (1+n_b+n_{\\hat{b}})z - n_c = 0$ encodes the admissible $\\theta$-range in the $\\lambda=0$ case.","core_discovery":"For the degenerate operator $L_p u = a_0 u_t - x_d^2 D_i(a_{ij} D_j u) + x_d b_i D_i u + x_d D_i(\\hat{b}_i u) + c u$ on $(-\\infty,T)\\times\\mathbb{R}^d_+$ and its elliptic analogue $L_e u = -x_d^2 D_i(a_{ij} D_j u) + x_d b_i D_i u + x_d D_i(\\hat{b}_i u) + c u$ on $\\mathbb{R}^d_+$, the paper proves that, under small partial-BMO assumptions on the coefficients, the equation $L u + \\lambda c_0 u = D_i F + f$ admits a unique weak solution in the weighted mixed-norm Sobolev space $H^1_{q,p,\\theta,\\omega}(T)$, with the bound $(1+\\sqrt{\\lambda})\\|u\\| + \\|x_d D_x u\\| \\leq N(\\|x_d^{-1}F\\| + (1+\\sqrt{\\lambda})^{-1}\\|f\\|)$. In the $\\lambda = 0$ regime the admissible weight exponent is $\\alpha p < \\theta < \\beta p$, where $\\alpha < \\beta$ are the two real roots of $z^2 + (1 + n_b + n_{\\hat{b}})z - n_c = 0$ and $n_b = b_d/a_{dd}$, $n_{\\hat{b}} = \\hat{b}_d/a_{dd}$, $n_c = c/a_{dd}$; this range is asserted to be optimal, with proof deferred. In one spatial dimension the elliptic range widens to all $\\theta$ except the two endpoints $\\alpha p$ and $\\beta p$.","pith_inferences":["If the announced optimality of $\\alpha p < \\theta < \\beta p$ is proved, it would show that in the undamped elliptic case the two endpoints are genuine thresholds: at $\\theta = \\alpha p$ or $\\theta = \\beta p$ some forcing in $L_{p,\\theta}$ would have no solution in $H^1_{p,\\theta}$, mirroring the one-dimensional explicit formula.","The dependence of Lemma 3.6 on the unproved nondegenerate estimate in Remark 3.7 is the point most worth checking; a failure there would not necessarily destroy the zeroth-order estimates but would break the higher-order localization and hence the main theorems.","Because the estimate is formulated with Muckenhoupt weights in time and mixed norms in space, the theory is ready-made for stochastic PDEs, where such spaces are standard, suggesting immediate applications to degenerate SPDEs with $x_d^2$-degenerate diffusion coefficients.","The change-of-variables technique used to diagonalize the leading coefficient is specific to the vanishing rate $x_d^2$; extending the method to $x_d^\\alpha$ with $\\alpha \\neq 2$ would require a different model space and likely a different set of admissible weights."],"forward_implications":["For any $\\lambda \\geq \\lambda_0$ sufficiently large, the parabolic theorem covers arbitrary $\\theta \\in \\mathbb{R}$, so the weight can be tuned to the behavior of the forcing without further restrictions on the coefficients.","At $\\lambda = 0$, the open interval $\\alpha p < \\theta < \\beta p$ is the stated admissible range; in one space dimension the elliptic result widens to all $\\theta$ except the two endpoints.","The Cauchy problem with zero initial data is solvable for every $\\lambda \\geq 0$ and every $\\theta \\in \\mathbb{R}$, with the constant allowed to depend on the length of the time interval.","The coefficient class includes functions measurable in $(t,x_d)$ with small tangential BMO, a substantially larger class than the uniformly continuous or constant-coefficient settings treated earlier.","No boundary condition is imposed; the results concern minimal-assumption solvability, and boundary traces can be handled separately as indicated by prior work."],"supporting_citations":[{"why":"Supplies the weighted Sobolev space framework, the norm equivalence (2.2), and the localization argument used throughout.","marker":"[31]"},{"why":"Provides the nondegenerate $H^1_p$ estimate for simple coefficients with $\\omega=1$, invoked in Remark 3.7.","marker":"[6]"},{"why":"Extends that nondegenerate estimate to general $A_p$ weights and partially SMO coefficients as used in Remark 3.7.","marker":"[9]"},{"why":"Justifies that the change of variables for $a_0(x_d)$ preserves the partially BMO assumptions in Remark 3.7.","marker":"[5]"},{"why":"Provides the weighted Hardy--Littlewood maximal theorem, extrapolation theorem, and box solvability used in Lemma 3.8, Lemmas 4.1--4.2, and Theorem 4.5.","marker":"[8]"},{"why":"Supplies solvability of nondegenerate systems in cylindrical domains used in the existence proofs for simple coefficients.","marker":"[4]"},{"why":"Source of the weighted partition-of-unity lemma (Lemma 5.2) used to localize the a priori estimate.","marker":"[27]"},{"why":"Provides the reduction idea for the elliptic theorem and the Cauchy problem (following its Theorems 2.1 and 2.6).","marker":"[32]"},{"why":"Supplies the uniqueness claim used in the complex interpolation step of Theorem 4.5.","marker":"[33]"}],"fun_headline_variants":["Degenerate PDEs get unique solutions and sharp bounds","Weighted Sobolev well-posedness for degenerate coefficients","New estimates for boundary-degenerate parabolic equations","Solving degenerate elliptic/parabolic in weighted norms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The higher-order estimates depend on an unproved solvability result for the nondegenerate 'frozen' equation with time- or $x_d$-dependent $a_0$ and $c_0$; the paper only sketches the reduction in Remark 3.7, and if that estimate fails the localization argument and the main theorems lose their foundation.","fun_headline_variants_meta":{"raw":{"variants":["Degenerate PDEs get unique solutions and sharp bounds","Weighted Sobolev well-posedness for degenerate coefficients","New estimates for boundary-degenerate parabolic equations","Solving degenerate elliptic/parabolic in weighted norms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000124,"raw_usage":{"total_tokens":1130,"prompt_tokens":997,"completion_tokens":133,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":71}},"tokens_in":613,"tokens_out":133,"duration_ms":2515,"temperature":1.0,"reasoning_tokens":71,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:00:45.207959+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to verify the unproved nondegenerate estimate from Remark 3.7 by testing $a_0(x_d) = 1 + \\varepsilon \\sin(x_d)$, $c_0 = 1$, $a_{ij} = \\delta_{ij}$ on the half-space: if the $H^1_{p,\\omega}$ a priori bound fails with constants independent of $\\varepsilon$ for some $p \\in (1,\\infty)$ and $A_p$ weight $\\omega$, then Lemma 3.6 and the main theorems lose their foundation; for the optimality claim at $\\lambda=0$ ($d \\geq 2$), one can test whether the estimate (2.11) holds for some $\\theta$ outside $\\alpha p < \\theta < \\beta p$ using constant-coefficient operators with non-radial forcing, since the one-dimensional explicit solution (6.4) identifies the two endpoints as the only singular exponents.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the nondegenerate $H^1_p$ estimate for simple coefficients with $\\omega=1$, invoked in Remark 3.7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends that nondegenerate estimate to general $A_p$ weights and partially SMO coefficients as used in Remark 3.7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Justifies that the change of variables for $a_0(x_d)$ preserves the partially BMO assumptions in Remark 3.7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the weighted Hardy--Littlewood maximal theorem, extrapolation theorem, and box solvability used in Lemma 3.8, Lemmas 4.1--4.2, and Theorem 4.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies solvability of nondegenerate systems in cylindrical domains used in the existence proofs for simple coefficients."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the weighted partition-of-unity lemma (Lemma 5.2) used to localize the a priori estimate."},{"cited_title":"Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Grad","cited_arxiv_id":null,"evidence_quote":"Supplies the uniqueness claim used in the complex interpolation step of Theorem 4.5."}],"review_version":1}